This is an ongoing series intended to help aspiring young (or young at heart) learners make the transition from the computation-based mathematics taught in US public schools and proof-based mathematics found at higher levels of mathematics education.
If you’re reading this, you probably like math to some extent. Or someone is making you read it, but let’s be optimistic and assume the former! Before we get into the nitty-gritty, though, you should probably know WHAT you’re reading.
What is this series about?
The goal of this series is to help aspiring mathematicians (or anyone looking to broaden their mathematical horizons) understand the difference between computationally-based math (arithmetic, middle-school algebra, calculus, etc.) and proof-based math (abstract algebra, topology, mathematical logic, etc.) Mathematics undergoes a paradigm shift some time after the calculus sequence: it’s no longer about applying algorithms that you’ve been told to memorize to compute some number of formula. Rather, it asks you to use your knowledge of mathematical facts, properties, theorems, etc. to discover more facts, properties, theorems, etc. The difference between the two is palpable, and if you’re anything like me the transition can be a bit overwhelming, especially if you’re studying on your own. This series aims to provide some example and tips to better illustrate how to think about proofs.
Who is this for?
The target audience is anyone who would like to have a better understanding of the flow of higher level mathematics, a feel for what it’s all about so to speak. The amount of mathematical sophistication required to understand each entry may vary, but a serious attempt will be made to make each one accessible to anyone regardless of mathematical skill.
If you struggle with the basics, like arithmetic or high school algebra, it might be prudent to refresh your knowledge in these areas first: mathematics is notorious for building on itself, and trying to learn higher level concepts without a solid understanding of the basics is like trying to play a game of Jenga with hooves instead of hands.
If you’re already an accomplished “prover,” i.e. you’ve taken a proof-based mathematics course at the university level and passed, then this series might move a bit too slowly for you.
If you’re looking for help understanding a specific mathematical concept, such as differentiation, the quadratic equation, ideals of rings, and so on, then this probably won’t help with that. Examples will be used to illustrate our talking points, but the main focus won’t be on understanding specific techniques or notions.
If you’re still here, and those disclaimers didn’t drive you away, then great! In the next entry we’ll get right into the thick of things, learning how to think about definitions and proofs in a mathematical sense.